In this paper we establish a number of implications between various
qualitative and quantitative versions of the global regularity problem for
the Navier–Stokes equations in the periodic, smooth finite energy, smooth
, Schwartz,
and mild
categories, and with or without a forcing term. In particular, we show that if one has global
well-posedness in
for the periodic Navier–Stokes problem with a forcing term, then one can obtain
global regularity both for periodic and for Schwartz initial data (thus yielding a
positive answer to both official formulations of the problem for the Clay Millennium
Prize), and can also obtain global almost smooth solutions from smooth
data
or smooth finite energy data, although we show in this category that fully smooth
solutions are not always possible. Our main new tools are localised energy and
enstrophy estimates to the Navier–Stokes equation that are applicable for large data
or long times, and which may be of independent interest.
